Inconsistency lemmas in algebraic logic

Show simple item record Raftery, James G. 2015-01-14T08:45:16Z 2015-01-14T08:45:16Z 2013-11
dc.description.abstract In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system is algebraized by a variety K, then has an inconsistency lemma—in the abstract sense—iff every algebra in K has a dually pseudo-complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) has a classical inconsistency lemma; (2) has a greatest compact theory and K is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in K form a Boolean lattice; (4) the compact congruences of any A ∈ K constitute a Boolean sublattice of the full congruence lattice of A. These results extend to quasivarieties and relative congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction-detachment theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The converses are false. en_ZA
dc.description.librarian hb2015 en_ZA
dc.description.uri en_ZA
dc.identifier.citation Raftery, JG 2013, 'Inconsistency lemmas in algebraic logic', Mathematical Logic Quarterly, vol. 59, no. 6, pp. 393-406. en_ZA
dc.identifier.issn 0942-5616 (print)
dc.identifier.issn 1521-3870 (online)
dc.identifier.other 10.1002/malq.201200020
dc.language.iso en en_ZA
dc.publisher Wiley en_ZA
dc.rights © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. This is the pre-peer reviewed version of the following article : Inconsistency lemmas in algebraic logic, Mathematical Logic Quarterly, vol. 59, no. 6, pp. 393-406, 2013, doi :10.1002/malq.201200020. The definite version is available at : en_ZA
dc.subject Deductive system en_ZA
dc.subject Inconsistency lemma en_ZA
dc.subject Protoalgebraic logic en_ZA
dc.subject Deduction-detachment theorem en_ZA
dc.subject Algebraizable logic en_ZA
dc.subject Pseudo-complement en_ZA
dc.subject Filtral variety en_ZA
dc.title Inconsistency lemmas in algebraic logic en_ZA
dc.type Postprint Article en_ZA

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